2025 MAST30026 Metric and Hilbert Spaces

Many administrative details about the organisation of the subject can (as always) be found in the subject handbook entry.

Consultation hours

See below under Exam preparation for the pre-exam consultation hours.

Lecture notes

Here is the latest version of the lecture notes (updated: 24 October @ 2pm).

Exercises

Here is the latest version of the exercises and solutions (updated: 24 October @ 2pm).

Tutorials

Tutorial classes start in week 2 of the semester. The tutorial sheets appear here at the start of the corresponding week, and solutions appear at the end of the week.

Assignments

The three assignments are posted both here and on the subject's Canvas page. Your solutions should be submitted via Canvas and Gradescope.

Assignment 1 (worth 4% of final mark)

Here is the pdf file for the first assignment, due 22 August at 20:00. Please read the cover page carefully.

The first assignment is a bit different because it is focussed on some of the prerequisite knowledge for this subject. The topics involved have been covered in previous subjects, but not necessarily with the same notation and terminology or in the way we will need this semester. The purpose of this assignment is to bring everyone up to speed; you may find the Prerequisites appendix of the lecture notes useful.

Ed Discussion post with instructions/FAQ/errata for the first assignment.

Solutions for the first assignment.

Assignment 2 (worth 8% of final mark)

Here is the pdf file for the second assignment, due 19 September at 20:00.

Solutions for the second assignment.

Assignment 3 (worth 8% of final mark)

Here is the (corrected on 27 September) pdf file for the third assignment, due 17 October at 20:00.

Solutions for the third assignment (revised on 25 October @ 7pm).

Special consideration

I suggest that you familiarise yourself with the University's special consideration procedures, which have been recently overhauled.

The relevant mechanism for this subject's assignments is a short extension, which must be requested before the assignment deadline. Short means ⩽ 5 business days. You can see the rules and apply online for a short extension. (Make sure to find the form relevant to the Faculty of Science.)

If your situation is not eligible for a short extension (e.g. if the assignment deadline has passed, or an extension of 5 days would be insufficient), you may apply for special consideration. If your application is successful, this would result in waiving the assignment in question (so that the rest of the assignments are re-weighted to make up the total of 20%).

Exam preparation

Pre-exam consultation hours

  • Tuesday 28 October 10:30am-12pm in Peter Hall room 107
  • Thursday 30 October 10:30am-12pm in the Alison Harcourt Seminar Room (Peter Hall room 162)
  • Wednesday 5 November 1pm-2pm in Peter Hall room 107
  • Friday 7 November 12pm-1:30pm in Peter Hall room 107

There will also be a consultation hour run by Chengjing and dedicated to assignment feedback, especially for assignment 3, on Thursday 6 November 10am-11am in the Alison Harcourt Seminar Room (Peter Hall room 162).

Everything you need to prepare for the exam is contained in the lecture notes, in the exercise booklet, in the tutorial sheets, and in the assignments. I have marked with a star (*) the sections in the lecture notes and the questions in the exercise booklet that can be considered highly optional (aka can be ignored) for the purposes of review of the material and the preparation for the exam.

Caveat: when looking at past exams, remember that "past performance is not indicative of future results".

In particular, note that this year's exam will have a slightly different format, with 9 "short" questions worth between 5 and 7 marks each for a total of 55 marks, followed by 1 "long" question worth 25 marks. "Short" means that you should typically be able to fit your solution on the first page of the question; you have a second page available if you need it, and some extra pages at the end, but if you consistently need that much space then you are probably being too verbose. Aim for short but complete and rigorous. In all the questions, but especially in the "long" one, if you cannot solve one part does not mean that you should skip the following parts as well; the parts appear in a logical progression, not necessarily of increasing difficulty.

Here is the 2024 final exam: questions and solutions.

Here is the 2023 final exam: questions and solutions.

Ed discussion board

Please see the subject's Canvas page for access to the discussion board.

Lecture recordings

Please see the subject's Canvas page for access to the lecture recordings.

Other references: prerequisite knowledge

The main prerequisites for the subject are the University of Melbourne's MAST20022 Group Theory and Linear Algebra and MAST20026 Real Analysis (or some equivalent subject, see the handbook entry for details).

For those of you arriving with a different background, this means a solid understanding of linear algebra and previous exposure to abstract algebra concepts like groups, group actions, fields; also required is a firm grasp of analysis of functions on the real line.

There are many excellent abstract algebra and real analysis texts out there, so feel free to grab some to use as a reference while working on this subject.

Here are some suggestions:

Other references: metric and Hilbert spaces

There are also many excellent analysis texts out there covering various of the topics we are studying. I'll list here any that I refer to.

Note: Many of these references may be accessible via the library system either as electronic resources or physical tomes.